Ask a New Question
Menu

apply the properties of integer exponent to generate equivalent expressions to (7^3)^7 times 7^20 divided by 6^-2 with only positive exponents, then solve the expression. provide the answer as an improper fraction

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
1 year ago

To apply the properties of integer exponents, we need to simplify each term individually and then combine them.

Let's break down each term and simplify:

First, simplify (7^3)^7 as 7^(3*7) = 7^21.
Next, simplify 7^20 as (7^2)^10 = 49^10.
Lastly, simplify 6^-2 as 1/(6^2) = 1/36.

Now, combine the simplified terms:
(7^21)*(49^10)/(1/36)

To multiply the terms, add the exponents of like bases:
7^21 * (7^2)^10 * 49^10 / (1/36)
= 7^21 * 7^20 * 49^10 / (1/36)
= 7^(21+20) * 49^10 / (1/36)
= 7^41 * 49^10 / (1/36).

To divide, subtract the exponents of like bases:
7^41 * 49^10 / (1/36)
= (7^41 * 49^10) * 36.

Now, let's evaluate this expression:
7^41 * 49^10 * 36
= (2^3)^41 * (7^2)^10 * 36
= 2^(3*41) * 7^(2*10) * 36
= 2^123 * 7^20 * 36.

Therefore, the equivalent expression with only positive exponents is 2^123 * 7^20 * 36.

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

To apply the properties of integer exponents and simplify the expression, let's break it down step-by-step:

Step 1: Simplify (7^3)^7
According to the property of integer exponents, when we raise a power to another power, we multiply the exponents. Therefore,
(7^3)^7 = 7^(3*7) = 7^21

Step 2: Simplify 7^20
Since 7^21 is multiplied by 7^20, we can combine them using the property of multiplying powers with the same base by adding the exponents. Hence,
7^21 times 7^20 = 7^(21+20) = 7^41

Step 3: Simplify 6^-2
Applying the property of a negative exponent, we know that a^-n is equal to 1/a^n. Consequently,
6^-2 = 1/6^2 = 1/36

Step 4: Combine the expressions
Now that we have simplified (7^3)^7, 7^20, and 6^-2, we can substitute back into the original expression and solve it:
(7^3)^7 times 7^20 divided by 6^-2 = 7^41 divided by 1/36

When we divide by a fraction, we multiply by its reciprocal. Thus,
7^41 divided by 1/36 = 7^41 times 36/1

Finally, applying the property of multiplying powers with the same base, we have:
7^41 times 36/1 = (7^41)*(36) = 7^41 * 36

Therefore, the simplified expression is 7^41 * 36.

However, it is not possible to provide this answer as an improper fraction since it is not a fraction.

User Icon for Explain Bot Explain Bot answered
1 year ago

To generate equivalent expressions using the properties of integer exponents, we can simplify the given expression step by step:

Step 1: Apply the power of a power property
(7^3)^7 = 7^(3*7) = 7^21

Step 2: Apply the power of a product property
7^20 = 7^(3*7) = (7^3)^7

Step 3: Apply the quotient of powers property
6^-2 = 1 / 6^2

Now, the simplified expression is:
(7^21 * 7^20) / (1 / 6^2)

Step 4: The product of powers property
7^21 * 7^20 = 7^(21+20) = 7^41

Step 5: Apply the reciprocal property
1 / 6^2 = 6^(-2)

The expression is now simplified to:
7^41 / 6^(-2)

Step 6: Apply the quotient of powers property
7^41 / 6^(-2) = 7^41 * 6^2

Now we can evaluate the expression:
7^41 = 1,323,415,773,484,786,885,835,434,698,785,666,976,737
6^2 = 36

So the value of the expression is:
1,323,415,773,484,786,885,835,434,698,785,666,976,737 * 36 = 47,563,155,244,254,757,488,078,446,358,917,604,748,532,532.

Therefore, the answer as an improper fraction is:
47563155244254757488078446358917604748532532/1